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May 25th

May 25th

# Fractions

• Definitions Before we can talk about fractions, we need to make sure that we
know some basic definitions:

- The top number in a fraction is called the numerator.

- The bottom number in a fraction is called the denominator. Remember
that if there isn't a denominator, the denominator is one !!

- In a proper fraction the numerator < denominator. So, in an improper
fraction, the numerator ≥ denominator.

- A mixed number is an integer and a proper fraction written together as
a single number.

- The reciprocal of a fraction is the fraction flipped over, i.e. the denomi-
nator over the numerator.

- A complex fraction is a fraction with a fraction in the numerator and/or
the denominator
.

• Improper Fractions and Mixed Numbers

- To convert an improper fraction to a mixed number:
1. Divide the denominator into the numerator.
2. The integer part of the mixed number is the result of the division.
3. The fraction part of the mixed number is remainder over the denomi-
nator.

Example 1 Convert to a mixed number.

1. 13 ÷ 6 = 2 with a remainder of 1.

2. The integer part is 2.
3. The fraction part is .
So,

- To convert a mixed number to an improper fraction:

1. Multiply the integer part by the denominator and add to the numer -
ator. This is the new numerator.

2. The improper fraction is the new numerator over the denominator.

Example 2 Convert to an improper fraction.

1. 5 × 3 + 2 = 17 is the new numerator.
2. The improper fraction is .
So, .

• Reducing a Fraction to Lowest Terms

1. Completely factor the numerator and denominator.

2. Cancel any common factors .

3. Once there are no common factors left, then the fraction is in lowest terms.

Example 3 Put in lowest terms

2. Since there are two 2s and one 3 in both the numerator and the denomi-
nator, these cancel, and we are left with .

3.The fraction in lowest terms is .
So, .

• Multiplying Fractions

1. Multiply the two numerators to get the new numerator.

2. Multiply the two denominators to get the new denominator.

3. The product of the two fractions is the new numerator over the new de-
nominator.

4. Reduce the fraction to lowest terms.

1. 3 × 5 = 15 is the new numerator.
2. 10 × 2 = 20 is the new denominator.
3.
4. Both the numerator and denominator have a factor of 5. Canceling this
out, we have that in lowest terms is .
So, .

• Dividing Fractions

1. Find the reciprocal of the second fraction.
2. Multiply the first fraction by the reciprocal of the second fraction.

1. The reciprocal of is .

2. By multiplying and then canceling the common factor of 3, we have:

So,

• Finding a Common Denominator

The common denominator of a set of fractions is a number which is divisible
by the denominators of all the fractions. The easiest way of finding a common
denominator is to multiply all the denominators together.

Example 6 Find a common denominator of and .

9 · 10 · 6 = 540 is a common denominator of the three fractions.

You may want to find the least common denominator, which is the smallest
number that is a common denominator of the fractions. This will require more
work to do, but, in a longer problem, it may make the rest of the problem easier.

1. Completely factor all denominators.

2. For each number that occurs in the factorizations, find the maximum num-
ber of times that number occurs in the factorization of a single denomina-
tor.

3. The factorization of the least common denominator must contain each
number that occurs in the factorizations, and they must be included the
number of times determined in Step 2.

Example 7 Find the least common denominator of and .

1. The denominators factor as follows: 9 = 3 · 3, 10 = 2 · 5, and 6 = 2 · 3.

2. 2 and 5 occur at most once in any denominator, but 3 occurs twice in the
factorization of 9.

3. The factorization of the least common denominator must contain one 2,
one 5, and two threes: 2 · 5 · 3 · 3 = 90.

So, the least common denominator of and is 90.

• Rewriting Fractions with a Common Denominator

1. Find a common denominator of the fractions.

2. For each fraction:
(a) Divide the common denominator by the denominator of the fraction.
(b) Multiply the numerator and the denominator by the result of the di-
vision. (Note that this does not change the value of the fraction since
it is really multiplication by one.)

Example 8 Rewrite and with a common denominator.

1. As seen in Example 7, the least common denominator is 90.

2. For each fraction:

So, and can be rewritten as and .

1. Rewrite the fractions with a common denominator.

2. The sum of the two fractions is the sum of the numerators over the com-
mon denominator. The difference of the two fractions is the difference of
the numerators over the common denominator.

3. Reduce the answer to lowest terms.

1. The least common denominator is 2 · 2 · 5 = 20, so we get and

3. is in lowest terms.

So,

Example 10 Subtract

1. The least common denominator is 7 · 3 = 21, so we get and

2. Subtracting, we get

3. is in lowest terms.

So,

• Simplifying Complex Fractions

1. Simplify the numerator and the denominator following the normal order
of operations .

2. Multiply the numerator by the reciprocal of the denominator.

Example 11 Simplify the complex fraction

1. Simplifying the numerator and denominator, we have:

2. Multiplying the numerator by the reciprocal of the denominator, we have:

3. Reducing to lowest terms, we have:

So,

• Getting Rid of Fractions in an Equation

1. Write every term as a fraction.

2. Get rid of all grouping symbols, such as parentheses, by multiplying.

3. Multiply every term on both sides of the equation by a common denomi-
nator of all the fractions.

4. If you simplify each term, all the fractions will be gone! Now proceed in
solving the equation .

Example 12 Solve for x:

1. Writing each term as a fraction, we have:
2. Removing the grouping symbols, we have:

3. The least common denominator of the fractions in the equation is 30. Mul-
tiply both sides by 30, we have:

4. Reducing the fractions, we have: 60x + 5 = 12 + 2x Solving the equation,
we have:

So, the solution to is

• Practice, Practice, Practice!!

1. Simplify:

2. Simplify:

3. Simplify:

4. Simplify:

5. Simplify:

6. Simplify:

7. Simplify:

8. Solve for x: